Can you accomplish this in theory? We found the Higgs boson, but can we remove it from particles?
There are two separate questions here. If we take the second part of your question first:
to make it have no mass?
Only a small percentage of the mass of an atom is due to the Higgs coupling. Most of it originates from the quark binding energy of the protons and neutrons nucleus. So even if we could somehow turn off the Higgs field we wouldn't change the mass of an atom very much. It would make the electrons massless, which wouldn't be great for the stability of any affected objects, but it wouldn't change the total mass very much as most of the mass is in the protons and neutrons.
On to the first part:
Is it possible to remove the Higgs Field
As far as we know there is no way to screen out the Higgs field or change the couplings of the elementary particles to the Higgs field. It is possible to make particles massless by heating your system enough to restore the broken Electro-Weak symmetry, but since this requires heating matter to 1,000,000,000,000,000 degrees we aren't going to be doing this on a macroscopic scale any time soon.